The only way I know to speak of `group-completion' of a symmetric monoidal $\infty$-category involves not only adjoining inverses of all objects but also inverses of all morphisms. In that sense, group-completion corresponds to first taking the classifying space (rather than the core) and then group-completing. For example, if $\mathcal{C}$ has an initial or terminal object, its classifying space is trivial, so its group completion is trivial.
Of course there are plenty of decent symmetric monoidal categories where every object has an inverse but which are not groupoids (for example, the full subcategory of $\text{Mod}_R$ on invertible modules). But maybe this does not deserve to be called `group-complete' because the inverse is not functorial (at least, not covariantly).
If the inverse is functorial (covariantly), then the category is a groupoid. Specifically, each morphism $f:X\rightarrow Y$ has an inverse (up to equivalence) $X\otimes Y\otimes(-f):Y\rightarrow X$.
Thus group-complete symmetric monoidal $\infty$-categories are exactly grouplike symmetric monoidal $\infty$-groupoids, also known as grouplike $\mathbb{E}_\infty$-spaces or connective spectra. The universal example of one of these is the sphere spectrum $\mathbb{S}$. Modules over $\mathbb{S}$ (in the sense of modules over a semiring $\infty$-category) are precisely the connective spectra, and the group-completion operation you want is $$-\otimes\mathbb{S}:\text{SymMon}\rightarrow\text{Mod}_\mathbb{S}\cong\text{Sp}_{\geq 0},$$ analogous to $$-\otimes\mathbb{Z}:\text{ComMon}\rightarrow\text{Mod}_\mathbb{Z}\cong\text{Ab}.$$ I discuss all of this in section 4 of my paper https://arxiv.org/abs/1606.05606